An interpolation theorem

Bulletin of Symbolic Logic 6 (4):447-462 (2000)
Abstract
Lyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of first-order logic, there is an interpolant in which each relation symbol appears positively (negatively) only if it appears positively (negatively) in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some fixed tuple U of unary predicates U, all formulae under consideration have all quantifiers explicitly relativised to one of the U. Under this stipulation, existential (universal) quantification over U contributes a positive (negative) occurrence of U. It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unifies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on positive vs. negative occurrences of predicates and on existentially vs. universally quantified sorts
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DOI 10.2307/420966
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References found in this work BETA
A New Look at the Interpolation Problem.Jacques Stern - 1975 - Journal of Symbolic Logic 40 (1):1-13.
Information Transfer Across Chu Spaces.J. Benthem - 2000 - Logic Journal of the IGPL 8 (6):719-731.

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Craig Interpolation for Networks of Sentences.H. Jerome Keisler & Jeffrey M. Keisler - 2012 - Annals of Pure and Applied Logic 163 (9):1322-1344.

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